Statistics- The interval is called a confidence interval and the spread

The interval is called a
confidence interval and the spread of the interval is dependent in part on the
spread of the sample statistic around the true population parameter. We know
how sample proportions vary around the true population proportion from last
lesson when we introduced the sampling distribution of a sample proportion. The
spread of our sampling distribution depended on the population proportion value
(p) and the sample size (n) (i.e. the standard deviation of the sampling
distribution was derived using a formula that used the population proportion
and sample size). However, now we do not know the population proportion (if we
did, then we would not have to use our sample value to make an inference about
the population value). We substitute the sample proportion value into our
formula for the standard deviation of the sampling distribution and now call
this measure of spread the standard error of the sample proportion.

Standard Error of the Sample
Proportion (SEP) = estimated standard deviation of the sample proportion
sampling distribution = √(sample proportion*(1-sample proportion)/sample size)

1. A random sample of 100
freshmen students at University Park campus is taken to estimate the percentage
of freshmen that have taken a tour of Pattee Library during orientation
week. Of the 100 freshmen, 40 (40%) of
them have taken a tour of Pattee Library.
The standard error (SE) for a sample percentage in this situation is
estimated to be approximately:
A) 4.0%
B) 4.9%
C) 0.24%
D) 24%

2. A random sample of 400
freshmen students at University Park campus is taken to estimate the percentage
of freshmen who have taken a tour of Pattee Library during orientation
week. Of the 400 freshmen, 160 (40%) of
them have taken a tour of Pattee Library.
The standard error (SE) for a sample percentage in this situation is
estimated to be approximately:
A) 4.9%
B) 4.0%
C) 2.45%
D) 24%

3. When the sample size
increases by 4 times (going from a sample of size 100 to a sample of size 400
with everything else remaining the same), the SE of the sampling distribution
for our sample proportion is:
A) ½ as large as it was before
the sample size increase
B) 2 times larger than it was
before the sample size increase
C) 4 times larger than it was
before the sample size increase
D) ¼ as large as it was before
the sample size increase

4. As sample size increases:
A) the SE increases
B) the SE remains the same
C) the SE decreases

Use the information below to
answer the questions that follow:
The confidence interval for
our true population proportion (for large random samples) is given by:

sample proportion +/- z*SEP

Where z* is a multiplier of
the SEP and the z value used depends on the level of confidence (see Table 10.1
in the online notes for z values for a particular level of confidence).

5. A random sample of 400
freshmen students at University Park campus is taken to estimate the percentage
of freshmen who have taken a tour of Pattee Library during orientation
week. Of the 400 freshmen, 160 (40%) of
them have taken a tour of Pattee Library.
The 95% confidence interval for the population percentage for all
freshmen who have taken a tour of Pattee Library during orientation week is:
A) 40% +/- 2.45%
B) 40% +/- 4.90%
C) 95% +/- 40%
D) 5% +/- 40%

6. An exit poll was done on
election night. The exit poll found that
204 out of 400 randomly selected voters (51%) had voted for the incumbent
Senator. The 95% confidence interval
for the population percentage of all voters who voted for the incumbent Senator
in that election is:
A) 51% +/- 5%
B) 51% +/- 2.5%
C) 95% +/- 5%
D) 95% +/- 2.5%

7. A random sample of 225
residents of a small mid-western town found that 80% approve of a new parking
ordinance. The 95% confidence interval
for the true population percentage of all residents who favor the parking
ordinance is:
A) 80% +/- 5.33%
B) 80% +/- 2.666%
C) 80% +/- 8%
D) 80% +/- 2%

8. Based on an exit poll done
on election night, the 95% confidence interval for the population percentage of
all voters who voted for the incumbent Senator was 52% +/- 3%. Based on this confidence interval, can we
state that we are 95% confident that the Senator got a majority of the vote
(greater than 50%) and was re-elected?
A) Yes
B) No

Use the information below to
answer the questions that follow:
The interval is called a
confidence interval and the spread of the interval is dependent in part on the
spread of the sample statistic around the true population parameter. We know
how sample means vary around the true population mean from last lesson when we
introduced the sampling distribution of a sample mean. The spread of our
sampling distribution depended on the population standard deviation value and
the sample size (i.e. the standard deviation of the sampling distribution was
derived using a formula that used the population standard deviation and sample
size). However, now we do not know the population value for our calculation. We
substitute the sample standard deviation value into our formula for the
standard deviation of the sampling distribution and now call this measure of
spread the standard error of the sample mean.

Standard Error of the Sample
Mean (SEM) = estimated standard deviation of the sample mean sampling
distribution = sample standard deviation/√n

9. A random sample of 81
residents of a Central Pennsylvania county found that the average restaurant
expenditure in the last month was $60 with a sample standard deviation of
$18. The standard error (SE) for the
average amount spent will be:
A) $18
B) $0.22
C) $2
D) $6.67

10. A random sample of 100
statistics students finds that the average time spent on an exam is 50 minutes
with a sample standard deviation of 5 minutes.
The standard error (SE) for the average time spent is:
A) 10.0 minutes
B) 1.0 minutes
C) 5.0 minutes
D) 0.5 minutes

11. A random sample of 400
statistics students finds that the average time spent on an exam is 50 minutes
with a sample standard deviation of 5 minutes.
The standard error (SE) for the average time spent is:
A) 0.5 minutes
B) 0.25 minutes
C) 0.0125 minutes
D) 5.0 minutes

12. Everything else being
equal, an increase in sample size from 100 to 400 will result in the standard
error (SE) for the average _________.
A) decreasing to ¼ of what it
was for the sample size of 100
B) decreasing to ½ of what it
was for the sample size of 100
C) increasing to 2 times what
it was for the sample size of 100
D) increasing to 4 times what
it was for the sample size of 100

Use the information below to answer
the questions that follow:
The confidence interval for
our true population mean (for large random samples, n > 30) is given by:

sample mean +/- z*SEM

Where z* is a multiplier of
the SEM that depends on the level of confidence (see Table 10.1 in the online
notes for z values for a particular level of confidence).

13. A random sample of 100
full time graduate students in the College of Science is taken to estimate the
amount spent on textbooks this semester.
The sampled students spent an average of $400 with a sample standard
deviation of $100. A 95% confidence
interval for the average amount spent by all full time graduate students in the
College of Science would be:
A) $400 +/- $100
B) $400 +/- $200
C) $400 +/- $10
D) $400 +/- $20

14. A random sample of 225
Penn State football season ticket holders is taken to estimate the population
average amount spent in State College on a typical home football game
weekend. The sample average was $500
with a sample standard deviation of $75.
A 95% confidence interval for the average amount spent by all Penn State
Football season ticket holders on a typical home football game weekend is:
A) $500 +/- $75
B) $500 +/- $5
C) $500 +/- $10
D) $500 +/- $150

15. A random sample of 81
Statistics undergraduate students found that the average amount of hours of
sleep per week during the past semester was 45.5 hours with a sample standard
deviation of 3 hours. A 95% confidence
interval for the average amount of hours of sleep per week by all Statistics
undergraduate students is:
A) 45.5 +/- 0.666 hours
B) 45.5 +/- 0.333 hours
C) 45.5 +/- 3 hours
D) 45.5 +/- 6 hours

16. A 95% confidence interval
for the average amount spent on textbooks by undergraduate students at a large
state university per semester is $400 +/- $50. Would it be correct to state
that the true population average could be $360?
A) Yes
B) No

We can also create a
confidence interval for the difference between two population proportions or
population means.
If we were to randomly choose
a sample from two different groups (independent samples) and then compare those
two sample values to one another and then do this many times with different
samples from each group, we would end up with a sampling distribution of the
DIFFERENCE between two sample values. This sampling distribution has a measure
of spread that is estimated by:

standard error of the
difference between two sample statistics (SED): = √((standard error of first
sample)^2 + (standard error of the second sample)^2)

We do not use the SED formula
when we have matched pairs (dependent samples). See example 10.7 in the online
notes.

17. A consumer testing
organization is comparing the battery life of two different models of PCs. They randomly sample 49 of each model of PC
(independent samples). The 49 PCs in
group 1 have an average battery life of 11 hours with a sample standard
deviation of 0.7 hours. The 49 PCs in
group 2 have an average battery life of 9 hours with a sample standard
deviation of 1.4 hours

The standard error for the
difference in the two averages is:
A) 0.7 + 1.4 = 2.1 hours
B) √(0.7^2 + 1.4^2) = 1.56
hours
C) √(0.1^2 + 0.2^2) = 0.223
hours
D) 0.1 + 0.2 = 0.3 hours

The confidence interval for
our true population difference is given by:
sample difference +/- z*SED

Where z* is a multiplier of
the SED that depends on the level of confidence (see Table 10.1 in the online
notes for z values for a particular level of confidence). When making
conclusions about our population values using a confidence interval of the
difference, we look to see if there is a ‘0’ within the interval. If there is a
‘0’ within the interval, we cannot conclude that there is a difference between
the two population values.

18. A consumer testing
organization is comparing the battery life of two different models of PCs. They randomly sample 49 of each model of
PC. The 49 PCs in group 1 have an
average battery life of 11 hours with a sample standard deviation of 0.7
hours. The 49 PCs in group 2 have an
average battery life of 9 hours with a sample standard deviation of 1.4 hours.

The 95% confidence interval of
the difference between the two group population means is:
A) (11-9) +/- 2 * 0.223 = 2
+/- 0.446 hours
B) (11-9) +/- 1 * 0.223 = 2
+/- 0.223 hours
C) (11-9) +/- 2 * 1.56 = 2 +/-
3.12 hours
D) (11-9) +/- 3 * 0.223 = 2
+/- 0.669 hours

19. As stated in the online
notes, holding everything else equal, a four fold increase in sample size alone
(as when the sample size increases from 100 to 400) will cut the margin of
error in half. Remembering that the margin of error = z*SE (the margin of error
is the distance on either side of the sample value that our confidence interval
covers), what is the reason that the margin of error is cut in half?

20. What happens to the width
of a confidence interval when the confidence level is increased from 90% to 95%
(holding all else constant)? Why? (Provide the reasoning behind your answer.)

21. We randomly select 1000
adults from a population of 2 million and also randomly select 1000 adults from
a population of 20 million. If the sample standard deviations are the same, how
will the margins of error compare for 95% confidence intervals for the true
population average? Will the margin of error for the sample from the larger
population be greater than, the same, or less than the margin of error for the
sample from the smaller population? Why? (Provide the reasoning behind your
answer.)

22. We have calculated a 95%
confidence interval for the difference between two population proportions. The
interval is 4% +/- 5%. Can we conclude that there is a difference between the
two group population proportions? Yes or No? Why? (Provide the reasoning behind
your answer.)

23. A group of college age
males and a group of college age females were sampled. and 95% confidence
intervals were created for the true population average of the amount spent on
pizza and related items in the past month. The male confidence interval is from
$40 to $60 ($50 +/- $10). The female confidence interval is from $25 to $35
($30 +/- $5). Is there a difference between the two group population mean
amounts spent? Why? Provide the reasoning behind your answer. (HINT: See page
10.4 in the online notes: “Statistical Significance and Confidence
Intervals.”)

 

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